10-28 R. F. WILLIAMS
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Abstract. The driving force of this paper is a local symmetry in lattices. The goal is two theorems: a partial converse to the Perron-Frobenius the- orem in dimension 3 and a characterization of conjugacy in Sl3(Z). In the process we develop a geometric approach to higher dimension con- tinued fractions, HDCF. HDCF is an active area with a long history: see for example Lagarias, [L],[Br]. The algorithm: Let Zr be the set of all lattice points within r > 0 of a ray L = {mP 2 Rn : m > 0} Let z1 denote the point in Zr closest to the origin. Having defined z1, .., zi, 1 i < n, let zi+1 be the point of Zr closest point into the origin, which is independent of z1, ...zi. Conjecture 1. z1, ..., zn is a basis of Zn. The proof of this conjecture in dimension n = 3 occupies the bulk of the paper. Here P 2 Rn is taken to have all positive components as usual, and we use a metric in L?. For further discussion and the relation of the conjecture to the Minkowski sequential minima theorem, see the section Remarks on the algorithm below. We have no such arithmetic theorems in dimension n > 3. However, in the first place, the symmetry theorem, its relation to arithmetic, and part of the bifurcation theorem are proved in all dimensions. Secondly, lots of computer studies in the next three dimensions have been done and we find no obstruction to this conjecture; in particular, for the partial converse mentioned above, we easily find bases of Zn, relative to which the matrix becomes positive, for n = 3, 4, 5, 6. A few examples are given below.

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